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Abstract We prove that in either the convergent or overconvergent setting, an absolutely irreducible $$F$$-isocrystal on the absolute product of two or more smooth schemes over perfect fields of characteristic $$p$$, further equipped with actions of the partial Frobenius maps, is an external product of $$F$$-isocrystals over the multiplicands. The corresponding statement for lisse $$\overline{{\mathbb{Q}}}_{\ell }$$-sheaves, for $$\ell \neq p$$ a prime, is a consequence of Drinfeld’s lemma on the fundamental groups of absolute products of schemes in characteristic $$p$$. The latter plays a key role in V. Lafforgue’s approach to the Langlands correspondence for reductive groups with $$\ell $$-adic coefficients; the $$p$$-adic analogue will be considered in subsequent work with Daxin Xu.
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This content will become publicly available on April 23, 2026
Tuning stiffness of mechanical metamaterial unit cells via transitions to second-order rigid and pre-stressed states
A design for a metamaterial with tunable stiffness is introduced. The material can be switched from floppy to rigid by changing the lengths of the constituent beams, which is demonstrated using a temperature-responsive hydrogel.
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- Award ID(s):
- 2247094
- PAR ID:
- 10584458
- Publisher / Repository:
- Royal Society of Chemistry
- Date Published:
- Journal Name:
- Soft Matter
- ISSN:
- 1744-683X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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